807 research outputs found
A lower bound on CNF encodings of the at-most-one constraint
Constraint "at most one" is a basic cardinality constraint which requires
that at most one of its boolean inputs is set to . This constraint is
widely used when translating a problem into a conjunctive normal form (CNF) and
we investigate its CNF encodings suitable for this purpose. An encoding differs
from a CNF representation of a function in that it can use auxiliary variables.
We are especially interested in propagation complete encodings which have the
property that unit propagation is strong enough to enforce consistency on input
variables. We show a lower bound on the number of clauses in any propagation
complete encoding of the "at most one" constraint. The lower bound almost
matches the size of the best known encodings. We also study an important case
of 2-CNF encodings where we show a slightly better lower bound. The lower bound
holds also for a related "exactly one" constraint.Comment: 38 pages, version 3 is significantly reorganized in order to improve
readabilit
On Tackling the Limits of Resolution in SAT Solving
The practical success of Boolean Satisfiability (SAT) solvers stems from the
CDCL (Conflict-Driven Clause Learning) approach to SAT solving. However, from a
propositional proof complexity perspective, CDCL is no more powerful than the
resolution proof system, for which many hard examples exist. This paper
proposes a new problem transformation, which enables reducing the decision
problem for formulas in conjunctive normal form (CNF) to the problem of solving
maximum satisfiability over Horn formulas. Given the new transformation, the
paper proves a polynomial bound on the number of MaxSAT resolution steps for
pigeonhole formulas. This result is in clear contrast with earlier results on
the length of proofs of MaxSAT resolution for pigeonhole formulas. The paper
also establishes the same polynomial bound in the case of modern core-guided
MaxSAT solvers. Experimental results, obtained on CNF formulas known to be hard
for CDCL SAT solvers, show that these can be efficiently solved with modern
MaxSAT solvers
Pseudo-Boolean Constraint Encodings for Conjunctive Normal Form and their Applications
In contrast to a single clause a pseudo-Boolean (PB) constraint is much more expressive and hence it is easier to define problems with the help of PB constraints. But while PB constraints provide us with a high-level problem description, it has been shown that solving PB constraints can be done faster with the help of a SAT solver. To apply such a solver to a PB constraint we have to encode it with clauses into conjunctive normal form (CNF). While we can find a basic encoding into CNF which is equivalent to a given PB constraint, the solving time of a SAT solver significantly depends on different properties of an encoding, e.g. the number of clauses or if generalized arc consistency (GAC) is maintained during the search for a solution.
There are various PB encodings that try to optimize or balance these properties. This thesis is about such encodings. For a better understanding of the research field an overview about the state-of-the art encodings is given. The focus of the overview is a simple but complete description of each encoding, such that any reader could use, implement and extent them in his own work. In addition two novel encodings are presented: The Sequential Weight Counter (SWC) encoding and the Binary Merger Encoding. While the SWC encoding provides a very simple structure – it is listed in four lines – empirical evaluation showed its practical usefulness in various applications. The Binary Merger encoding reduces the number of clauses a PB encoding needs while having the important GAC property. To the best of our knowledge currently no other encoding has a lower upper bound for the number of clauses produced by a PB encoding with this property. This is an important improvement of the state-of-the art, since both GAC and a low number of clauses are vital for an improved solving time of the SAT solver. The thesis also contributes to the development of new applications for PB constraint encodings. The programming library PBLib provides researchers with an open source implementation of almost all PB encodings – including the encodings for the special cases at-most-one and cardinality constraints. The PBLib is also the foundation of the presented weighted MaxSAT solver optimax, the PBO solver pbsolver and the WBO, PBO and weighted MaxSAT solver npSolver
Fuzzy Maximum Satisfiability
In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to
{\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the
problem of finding an assignment to the variables in {\Phi} that satisfies the
maximum number of formulae. Three possible solutions (encodings) are proposed
to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer
Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem
(WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have
numerous applications in optimization problems that involve vagueness.Comment: 10 page
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